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Giải bài 4.57 trang 73 sách bài tập toán 7 - Kết nối tri thức với cuộc sống

Cho tam giác ABC cân tại đỉnh A. Gọi M là trung điểm của BC. Trên cạnh AB và AC lấy các điểm P, Q sao cho MP, MQ lần lượt vuông góc với AB, AC (H.4.59) a)Chứng minh rằng MP = MQ và AP = AQ b) Đường thẳng PQ có vuông góc với AM không? Vì sao?

♓Tổng hợp đề thi học kì 2 lớp 7 tất cả các môn - Kết nối tri thức

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Đề bài

Cho tam giác ABC cân tại đỉnh A. Gọi M là trung điểm của BC. Trên cạnh AB và AC lấy các điểm P, Q sao cho MP, MQ lần lượt vuông góc với AB, AC (H.4.59) a) Chứng minh rằng MP = MQ và AP = AQ b) Đường thẳng PQ có vuông góc với AM không? Vì sao?

Phương pháp giải - Xem chi tiết

a) Chứng minh: \(\Delta PBM = \Delta QCM\left( {ch - gn} \right)\) b) Chứng minh: AM là đường trung trực của đoạn thẳng PQ

Lời giải chi tiết

a) Vì tam giác ABC cân tại A nên \(AB = AC; \widehat{ABC}=\widehat{ACB}\).Xét \(\Delta PBM\) và \(\Delta QCM\) có:\(\widehat {BPM} = \widehat {CQM} = {90^0}\)MB = MC (gt)\(\begin{array}{l}\widehat {MBP}= \widehat {MCQ} (cmt) \\ \Rightarrow \Delta PBM = \Delta QCM\left( {ch - gn} \right)\end{array}\)\( \Rightarrow MP = MQ\) (2 cạnh tương ứng)Ta có:\(\left\{ \begin{array}{l}AP = AB - BP\\AQ = AC - CQ\\AB=AC;BP=CP\end{array} \right. \Rightarrow AP = AQ\)b)Ta có: A, M cùng cách đều hai điểm P, Q nên AM là đường trung trực của đoạn thẳng PQ. Do đó AM vuông góc với PQ.

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